Chiral Rings Do Not Suffice: N=(2,2) Theories with Nonzero Fundamental Group

TL;DR

This paper explores the limitations of chiral rings in N=(2,2) theories with nontrivial fundamental groups, revealing ambiguities in the moduli space and implications for the Torelli problem in Calabi-Yau threefolds.

Contribution

It demonstrates that chiral rings do not fully determine the moduli space in certain non-simply-connected Calabi-Yau models, highlighting new ambiguities and counterexamples.

Findings

Chiral rings can be identical for different conformal field theories.

The ambiguity is explained via A-model and B-model perspectives.

Provides a counterexample to the global Torelli problem.

Abstract

The Kahler moduli space of a particular non-simply-connected Calabi-Yau manifold is mapped out using mirror symmetry. It is found that, for the model considered, the chiral ring may be identical for different associated conformal field theories. This ambiguity is explained in terms of both A-model and B-model language. It also provides an apparent counterexample to the global Torelli problem for Calabi-Yau threefolds.